## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 37, Number 1 (1966), 120-132.

### "Optimal" One-Sample Distribution-Free Tests and Their Two-Sample Extensions

#### Abstract

The object of this paper is the development of a theory of optimal one-sample goodness-of-fit tests and of optimal two-sample randomized distribution-free (DF) statistics analogous to the well-known results of Hoeffding (1951), Terry (1952), Lehmann (1953), (1959), Chernoff and Savage (1958), Capon (1961) and others for two-sample nonrandomized rank statistics. For $Y_1, \cdots, Y_n$ a random sample from a population with continuous distribution function $G$, one tests in the one-sample case $H_0 : G = F$ vs. $H_1 : G \neq F$, where $F$ is some known continuous distribution function. From the Neyman-Pearson lemma, distribution-free tests that are most powerful (MP) for any $H$ vs. $K$ satisfying $KH^{-1} = GF^{-1}$, are obtained. From these MP distribution-free tests, one can on paralleling the derivations ([14], [25], [17], [18], [7]) for locally MP tests in the two-sample case obtain locally MP tests in the one-sample case. Further, it is found that the class of alternatives, for which a critical region of the form $\lbrack\sum J\lbrack F(y_i)\rbrack > c\rbrack$ is locally MP, is the class of $G$'s that consists of "contaminated" Koopman-Pitman distributions as given in Section 5. Randomized versions of the two-sample MP and locally MP rank statistics are considered and shown to be asymptotically equivalent to the locally MP rank statistics.

#### Article information

**Source**

Ann. Math. Statist., Volume 37, Number 1 (1966), 120-132.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177699603

**Digital Object Identifier**

doi:10.1214/aoms/1177699603

**Mathematical Reviews number (MathSciNet)**

MR198621

**Zentralblatt MATH identifier**

0142.15703

**JSTOR**

links.jstor.org

#### Citation

Bell, C. B.; Doksum, K. A. "Optimal" One-Sample Distribution-Free Tests and Their Two-Sample Extensions. Ann. Math. Statist. 37 (1966), no. 1, 120--132. doi:10.1214/aoms/1177699603. https://projecteuclid.org/euclid.aoms/1177699603